How Traders Manage Their
Exposures
1
A Trader’s Gold Portfolio. How Should
Risks Be Hedged?
Position
Spot Gold
Forward Contracts
Futures Contracts
Swaps
Options
Exotics
Total
Value ($)
180,000
– 60,000
2,000
80,000
–110,000
25,000
117,000
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Delta
• Delta of a portfolio is the partial derivative of a
portfolio value with respect to the price of the
underlying asset (gold in this case)
Delta =
¶Õ
¶S
• Suppose that a $0.1 increase in the price of gold
leads to the gold portfolio decreasing in value by $100
• The delta of the portfolio is -1000
- 100
Delta =
0.1
3
delta neutral
• The portfolio could be hedged against
short-term changes in the price of gold by
buying 1000 ounces of gold. This is known
as making the portfolio delta neutral
• Delta neutral portfolio has zero delta.
4
Linear vs Nonlinear Products
• When the price of a product is linearly
dependent on the price of an underlying
asset, (e.g., forward contracts, futures
contracts and swaps) a ``hedge and
forget’’ strategy can be used (since delta
does not change at all)
• Non-linear products require the hedge to
be rebalanced to preserve delta neutrality
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Example
• A bank has sold for 100,000 European call
option on 100,000 shares of a nondividend
paying stock for $300,000.
• S0 = 49, K = 50, r = 5%, s = 20%,
T = 20 weeks, µ = 13%
• The Black-Scholes value of the option is
$240,000
• How does the bank hedge its risk to lock in a
$60,000 profit?
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Delta of the Option
Option
price
Slope = D
B
A
Stock price
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Delta Hedging
•
•
•
•
Initially the delta of the option is 0.522
The delta of the portfolio is -0.522*100,000
=-52,200
This means that 52,200 shares are
purchased to create a delta neutral position
• But, if a week later delta of the option
changes to 0.458, then the portfolio delta
changes to -45,800 and 6,400 shares must
be sold to maintain delta neutrality
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Option closes in the money
Week
Stock Price
Delta
Shares
Purchased
0
49.00
0.522
52,200
1
48.12
0.458
(6,400)
2
47.37
0.400
(5,800)
3
50.25
0.596
19,600
….
…..
….
…..
19
55.87
1.000
1,000
20
57.25
1.000
0
Gamma
• Gamma (G) is the rate of change of delta (D)
with respect to the price of the underlying
asset
¶ 2P
G=
¶S 2
• If Gamma is large in absolute term, then delta
is highly sensitive to the price of the underlying
asset
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Gamma Measures the Delta
Hedging Errors Caused By
Curvature
Call
price
C''
C'
C
Stock price
S
S'
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Gamma Neutral Portfolio
• A linear product has zero gamma.
• A delta neutral portfolio has a gamma equal to G
• To make this portfolio gamma neutral, the numberwT
of traded option added to the portfolio solves
wT GT + G = 0
• The Delta needs to be rebalanced
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Vega
• Vega (n) is the rate of change of the value of a
derivatives portfolio with respect to volatility
¶P
n=
¶s
• If the vega is high in absolute term, the portfolio
value is very sensitive to small changes in
volatility.
13
• To make this portfolio vega neutral, the number
of traded option added to the portfolio solves
wTn T +n = 0
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Example
Consider a portfolio that is delta neutral, with
a gamma of -5000 and a vega of -8000.
A traded option has a gamma of 0.5, a
vega of 2.0, and a delta of 0.6.
(1) How do you make the portfolio vega
neutral? As a result, what are the new
delta and gamma for the new portfolio?
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(2) Suppose that there is a second
traded option with a gamma of
0.8, a vega of 1.2, and a delta of
0.5.
How do you make the
portfolio both gamma and vega
neutral? What is the resulting new
delta? How to make the final
portfolio delta netrual as well?
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Solution
(1) -8000+w(2)=0,
w=4000 (long 4000 traded
options)
As a result, the new delta is
0.6*4000=2400 (sell or short 2400
units of the assets to maintain
delta neutrality)
The new gamma is
-5000+0.5*4000=-3000
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- 5,000 + 0.5w1 + 0.8w2 = 0
- 8,000 + 2.0w1 + 1.2 w2 = 0
w1 = 400
w2 = 6,000
400 ´ 0.6 + 6,000 ´ 0.5 = 3,240.
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The portfolio can therefore be made gamma
and vega neutral by including 400 of the
first traded option and 6000 of the second
traded option. The delta of the portfolio
after the addition of the positions in the two
traded options is 3240 Hence, 3240 units
of the asset would have to be sold to
maintain delta neutrality.
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Theta
• Theta (Q) of a derivative (or portfolio of
derivatives) is the rate of change of the value
with respect to the passage of time
• The theta of a call or put is usually negative.
This means that, if time passes with the price of
the underlying asset and its volatility remaining
the same, the value of the option declines
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Rho
• Rho is the partial derivative with respect to
to a parallel shift in all interest rates in a
particular country
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Taylor Series Expansion
(assume volatility and interest rates are
constant)
¶P
¶P
1¶ P
2
DP =
DS +
Dt +
(DS ) +
2
¶S
¶t
2 ¶S
2
2
1¶ P
¶ P
2
(Dt ) +
DSDt + !
2
2 ¶t
¶t¶S
2
22
¶P
¶P
1¶ P
2
DP =
DS +
Dt +
(DS ) +
2
¶S
¶t
2 ¶S
2
2
1¶ P
¶ P
2
(Dt ) +
DSDt + !
2
2 ¶t
¶t¶S
2
23
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Interpretation of Gamma
• For a delta neutral portfolio,
D » Q Dt + ½GDS 2
D
D
DS
DS
Positive Gamma
Negative Gamma
Taylor Series Expansion when Volatility is
change but interest rates are unchanged
¶P
¶P
¶P
1¶ P
2
DP =
DS +
Ds +
Dt +
(
D
S
)
2
¶S
¶s
¶t
2 ¶S
1 ¶ 2P
2
+
(Ds) +
2
2 ¶s
2
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Managing Delta, Gamma, &
Vega
• D can be changed by taking a position in
the underlying
• To adjust G & n it is necessary to take a
position in an option or other derivative
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Hedging in Practice
•
•
•
Traders usually ensure that their portfolios
are delta-neutral at least once a day
Whenever the opportunity arises, they
improve gamma and vega
A trader responsible for all trading involving
a particular asset must keep gamma and
vega within limits set by risk management
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